The choice of ω 1 and combinatorics at ℵ ω

Transcription

1 The choice of ω 1 and combinatorics at ℵ ω Spencer Unger UCLA October 19, 2013

2 Outline Combinatorial properties Theorems using supercompactness Bringing results down to ℵ ω Some new theorems A sketch of a proof Remarks

3 Combinatorial properties There are three weak square principles that we will be interested in ν, ν + I [ν + ] and All scales of length ν + are good. In fact they are strictly decreasing in strength by a theorem of Shelah and some consistency results. Theorem (Shelah) For a singular cardinal ν, ν implies ν + I [ν + ] implies All scales of length ν + are good.

4 Weak square Weak square was introduced by Jensen who proved that it holds if and only if there is a special Aronszajn tree. Definition Let ν be a cardinal. A ν-sequence is a sequence C α α < ν + such that 1. for all α < ν +, 1 C α ν, 2. for all α < ν + and for all C C α, C is club in α and o.t.(c) ν and 3. for all α < ν +, C C α and β lim(c), C β C β.

5 Approachability The notion of approachability was introduced by Shelah. In fact there is an ideal of approachable sets. Definition Let µ be a regular cardinal and x α α < µ be a sequence of bounded subsets of µ. An ordinal γ < µ is approachable with respect to x if there is A γ cofinal such that o.t.(a) = cf(γ) and for all β < γ, there is δ < γ such that A β = x δ. Definition Let µ be a cardinal. A set S µ is in the collection of approachable subsets I [µ] if and only if there are a club C µ and a sequence x such that for all γ S C, γ is approachable with respect to x. Fact For a regular cardinal µ, I [µ] is an ideal. If µ I [µ], then we say that µ has the Approachability property.

6 Scales Scales are a notion from Shelah s PCF theory. The general setting is a singular cardinal ν with an increasing and cofinal sequence ν i i < cf(ν) of regular cardinals less than ν. Given members f, g i ν i we say that f < g if and only if there is a j < cf(ν) such that for all i j, f (i) < g(i). A sequence of functions f α α < ν + is a scale of length ν + in i ν i if it is increasing and cofinal in i ν i under the ordering <. A point γ < ν + with cf(γ) > cf(ν) is good for a scale f of length ν + if there are A γ cofinal and j < cf(ν) such that for all i j the sequence f α (i) α A is strictly increasing.

7 Scales continued A scale f is good if there is a club C ν + such that all γ in C of cofinality greater than cf(ν) are good for f. A scale f is bad if it is not good. We say there is a bad scale of length ν + if there is a bad scale of length ν + in some product ν i. Recall Theorem (Shelah) For a singular ν, ν implies ν + I [ν + ] implies All scales of length ν + are good.

8 Stationary reflection Let µ be a regular cardinal. Recall that a stationary subset S of µ reflects if there is an α < µ such that S α is stationary.

9 Theorems from supercompactness It turns out that all of the weak square principles above fail in the presence of supercompactness. Theorem Let κ be a supercompact cardinal and ν > κ be singular with cf(ν) < κ, every scale of length ν + is bad. Let f α α < ν + be a scale in the product i ν i. Let j : V M witness that κ is ν + -supercompact. In particular crit(j) = κ and ν+ M M.

10 A proof continued Let γ = sup j ν + and note that γ < j(ν + ). It is not hard to show that the function, i sup j ν i is a so called exact upper bound for j(f ) α α < γ. It follows that γ is bad for j( f ). Standard reflection arguments show that there is a stationary set S ν + of bad points for f. Moreover S concentrates on cofinalities µ + where µ is a singular cardinal of cofinality cf(ν).

11 Theorems from supercompactness continued Theorem Suppose that ν is a singular limit of supercompact cardinals, then every stationary subset of ν + reflects. Proof. Suppose that S ν + is stationary. There is a regular cardinal δ < ν such that S = def S cof(δ) is stationary. It is enough to show that S reflects. Let κ be a supercompact cardinal greater than δ and let j : V M witness that κ is ν + -supercompact. Now since δ < κ we can show that j(s ) sup j ν + is stationary in M. It follows that S reflects in V.

12 Bringing results down to ℵ ω Theorem Assuming there is a supercompact cardinal it is relatively consistent that there is a bad scale of length ℵ ω+1, in particular ℵ ω fails. Ideas of the proof: Fix a scale of length κ +ω+1 in the product κ +n. By our previous argument the scale has stationarily many bad points of some cofinality µ + < κ. Force with Coll(ω, µ) Coll(µ ++, < κ). Show that the scale is still a scale. Show that the stationary set of bad points is preserved. Show that it is still a set of bad points, but now concentrating on cofinality ω 1.

13 Bringing results down continued Theorem (Magidor) Assume that there are infinitely many supercompact cardinals, then it is consistent that every stationary subset of ℵ ω+1 reflects. Ideas of the proof: Let κ n n < ω be an increasing sequence of supercompact cardinals. Iterate Levy Collapses with full support starting with Coll(ω < κ 0 ) and in general n Q n = Coll(κ n, < κ n+1 ). Try to repeat the proof we sketched above. Now that there is forcing involved, we must use generic elementary embeddings. The proof is mostly the same except that we need to force to see that the generic embeddings exist. The key point in the proof is to show that the forcing that adds some elementary embedding cannot destroy the stationarity of some set. To prove this fact we actually show that in the final model ℵ ω+1 I [ℵ ω+1 ].

14 The choice of ω 1 Some Remarks: In the proof that it is consistent that there is bad scale of length ℵ ω+1, we chose µ + which was a successor of a singular cardinal of cofinality ω to become ω 1. In the proof of Magidor s theorem above we made a supercompact cardinal into ω 1. Recall that by the theorem of Shelah at the beginning, there is a bad scale of length ℵ ω+1 is incompatible with the approachability property at ℵ ω+1. We will see later Magidor s theorem requires a formerly supercompact ω 1.

15 One more theorem for background Theorem (Cummings and Foreman) Assuming there are infinitely many supercompact cardinals, it is consistent that the tree property holds at ℵ n for 2 n < ω.

16 Some new theorems Let R ω be the Cummings-Foreman iteration. For each of the following theorems we assume that there are infinitely many supercompact cardinals. Theorem (U) Suppose that V is a model obtained by forcing with Coll(ω, < κ) for some supercompact κ, then in V [R ω ] 1. ℵ ω+1 I [ℵ ω+1 ] and 2. for every n < ω, every stationary subset of ℵ ω+1 cof(ℵ n ) reflects at a point of cofinality ℵ n+1. Theorem (U) There are a µ < κ 0 and a generic object d G ω for Coll(ω, µ) (R ω ) V [Coll(ω,µ)] such that in V [d G ω ], there are a bad scale of length ℵ ω+1 and a non-reflecting stationary subset of ℵ ω+1.

17 Another theorem Theorem (U and Fontanella independently) In the Cummings-Foreman model, ITP(ℵ n, λ) holds for all n with 1 < n < ω and all λ ℵ n.

18 An explanitory theorem Let κ n n < ω be an increasing sequence of supercompact cardinals. Let C be the direct sum of Coll(ω, µ) for µ < κ 0 a singular cardinal of cofinality ω. Let Ṙ be a C-name for the full support iteration of Levy collapses to make κ n into ℵ n+1 for all n < ω. Theorem There is a generic object c G for C Ṙ such that in V [c G] there are a bad scale of length ℵ ω+1 and a non reflecting stationary subset of ℵ ω+1.

19 Ideas from the proof Let V 0 be a model of GCH with infinitely many supercompact cardinals κ n for n < ω and let V be the extension of V 0 by the Laver preparation for κ 0. Let f be a scale in V 0 in κ n. Note that f is a bad scale in V 0 and in V. It is not hard to show that f remains a scale in V [C Ṙ]. It is also not hard to show if α is a bad point of f in V 0, then it is still a bad point of f in V [C Ṙ]. The key difficulty is to show that the stationary set of bad points from V 0 is preserved in the extension by some generic for C Ṙ. Recall that C chooses ω 1.

20 A useful lemma Lemma Suppose that V is obtained from V 0 by forcing with the Laver preparation for some supercompact cardinal κ. Further assume that f V 0 is a bad scale of length ν + for singular ν with cf(ν) < κ with set of bad points S. If P is κ-directed closed forcing which preserves ν +, then in V [P] S contains a stationary set of bad points.

21 Ideas continued To show that there is a non-reflecting stationary set we note that actually the stationary set of bad points of cofinality ω 1 does not reflect. To see this note that the set of bad points cannot reflect at a good point. Then we just show that all points of cofinality ℵ n for n > 1 are good for our scale. It follows that the set of bad points does not reflect.

22 Remarks Extra difficulties when the forcing is not just iterated Levy Collapses. The generic embeddings are added my much more complex forcing. It is no longer true that the forcing can be written as (κ n -cc forcing) * (κ n -closed forcing). We can replace κ n -closed with < κ n -distributive and this is enough.

23 The paper A model of Cummings and Foreman revisited, Submitted